Bandwidth, Sample Rate, and Nyquist Theorem

Table of Contents

1. Introduction

The two major components in a high-speed digitizer's analog front end are the analog input path and the analog-to-digital converter (ADC). The analog input path attenuates, amplifies, filters, and/or couples the signal to optimize the digitization by the ADC. The ADC samples the conditioned waveform and converts the analog input signal to digital values that represent the conditioned input signal.

Figure 1

Bandwidth describes the analog front end's ability to get a signal from the outside world to the ADC with minimal amplitude loss. Sample rate is the frequency at which the ADC converts the analog input waveform to digital data. The Nyquist Theorem explains the relationship between the sample rate and the frequency of the measured signal. Each of these terms is discussed in more detail below.

2. Bandwidth

Bandwidth describes the frequency range in which the input signal can pass through the analog front end with minimal amplitude loss - from the tip of the probe or test fixture to the input of the ADC. Bandwidth is specified as the frequency at which a sinusoidal input signal is attenuated to 70.7% of its original amplitude, also known as the -3 dB point. The following figure shows the typical input response for a 100 MHz high-speed digitizer.

Figure 2

For example, if you input a 1 V, 100 MHz sine wave into high-speed digitizer with a bandwidth of 100 MHz, the signal will be attenuated by the digitizer’s analog input path and the sampled waveform will have an amplitude of approximately 0.7 V.

Figure 3

It is recommended that the bandwidth of your digitizer be 3 to 5 times the highest frequency component of interest in the measured signal to capture the signal with minimal amplitude error (bandwidth required = (3 to 5)*frequency of interest). The theoretical amplitude error of a measured signal can be calculated from the ratio of the digitizer's bandwidth in relation to the input signal frequency (R).

Figure 4

For example, the error in amplitude when measuring a 50 MHz sinusoidal signal with a 100 MHz high-speed digitizer, which yields a ratio of R=2, is approximately 10.5%.

Another important topic related to bandwidth is rise time. The rise time of an input signal is the time for a signal to transition from 10% to 90% of the maximum signal amplitude and is inversely related to bandwidth by the following formula, based on the one pole model, R-C limited input response.

Figure 5

This means that the rise time of a 100 MHz digitizer input path is 3.5 ns. It is recommended that the rise time of the digitizer input path be 1/3 to 1/5 the rise time of the measured signal to capture the signal with minimal rise time error. The theoretical rise time measured (Trm) can be calculated from the rise time of the digitizer (Trd) and the actual rise time of the input signal (Trs).

Figure 6

For example, the rise time measurement when measuring a signal with 12 ns rise time with a 100 MHz digitizer is approximately 12.5 ns.

3. Sample Rate

Sample rate is not directly related to the bandwidth specifications of a high-speed digitizer. Sample rate is the speed at which the digitizer’s ADC converts the input signal, after the signal has passed through the analog input path, to digital values that represent the voltage level. This means that the digitizer will sample the signal after any attenuation, gain, and/or filtering has been applied by the analog input path, and convert the resulting waveform to digital representation. The sample rate of a high-speed digitizer is based on the sample clock that tells the ADC when to convert the instantaneous analog voltage to the digital values. National Instruments high-speed digitizers support a variable effective sample rate derived from the maximum sample rate of the device. For example, the NI 5112 has a maximum sample rate of 100 Megasamples/second (MS/s) and can be set to rates of (100MS/s)/n, where n = 1,2,3,4,....

4. Nyquist Theorem

The Nyquist theorem states that a signal must be sampled at a rate greater than twice the highest frequency component of the signal to accurately reconstruct the waveform; otherwise, the high-frequency content will alias at a frequency inside the spectrum of interest (passband). An alias is a false lower frequency component that appears in sampled data acquired at too low a sampling rate. The following figure shows a 5 MHz sine wave digitized by a 6 MS/s ADC. The dotted line indicates the aliased signal recorded by the ADC and is sampled as a 1 MHz signal instead of a 5 MHz signal.

Figure 8: Sine Wave Demonstrating the Nyquist Frequency

The 5 MHz frequency aliases back in the passband, falsely appearing as a 1 MHz sine wave. To prevent aliasing in the passband, you can use a lowpass filter to limit the frequency of the input signal or increase your sampling rate.